A basic approach to body pH

The physical distance between Europe and North America is perhaps a few thousand kilometers
but from a acid base point of view, we are talking light years. It is sad
that acid-base (which should be relatively easily understood, now
that we have modern computers to do the dirty work for us) is plagued by
unnecessary argument. Particularly vituperative comments have come from
Severinghaus
and
Siggaard-Andersen, criticising the work of Stewart
and the "strong ion difference" concept.
We believe that although Siggaard-Andersen has made a massive
contribution to the understanding of acid-base, Stewart has provided
a sound mathematical approach which is of great use in understanding
acid-base. This article briefly examines Peter Stewart's approach to
acid-base. It's best read carefully,
but you can jump straight to the calculator,
a Java applet that implements his model!

Basis

This web page makes few assumptions about your knowledge. It should
be easily understood by anyone with a high-school education and
a bit of a background in clinical medicine and physiology. We try to
keep things simple - if you feel that our tone is patronising, then you
are probably too advanced to be reading such simple stuff! Note that
a bit of spadework is still required to work through the maths - pen and
paper remain a wonderful medium for this! Because solutions for the
more complex equations require the use of computers, we use a tiny little
bit of "computerspeak" - as little as we can get away with!
We do however feel obliged to use a star
(*) to indicate multiplication and a slash (/) for division. This page does
not assume that you can speak a computer language.

Remember
from physiology that if you have two substances A and B reversibly
combining to form substance C, that when the reaction is at equilibrium
the equation governing the equilibrium is:

[A] * [B] = K * [C]

where K is the rate constant for the reaction.

Throughout the body pH is vitally important. Anyone who has a blood pH
of below 7.0 is likely to die soon if this isn't corrected. Likewise, a pH
of over 7.6 is frequently associated with serious illness and death.
Remember that pH is the log of the reciprocal of the hydrogen ion concentration.
Why use such a convoluted measure, and not just talk about the concentration
of hydrogen ions? The answer is mainly historical but also perhaps reflects
the ongoing inability of humans to cope with concepts that cannot be counted
on fingers (with perhaps a little help from the toes)! Rather talk about
pH 7 than 1 * 10-7 mol/litre, or (god forbid!) 100 nanomol/l of
hydrogen ion.

Unfortunately, such terminology has only served to complicate things,
especially when we start plotting how pH changes when we change another
ion concentration - we have to continually remember that we are dealing
with non-linear axes on any graph we might draw. We will try and clear
up some of the confusion. It goes without saying that in any solution
mass is conserved (the amount
of each component remains constant unless some of that substance is
added or removed, either physically or by participating in a chemical
reaction), and that any aqueous solution is
electrically neutral (the
number of positive ions equals the number of negative ones). But let's
review a few other fundamental definitions:

We now have two simultaneous equations, and it is easy to see that
[H+] = [OH-], so [H+] is the square
root of KW'. We can also see the practical significance
of our definition of an acid-base neutral solution
- if the hydrogen and hydroxyl ion concentrations are the same,
then

[H+] * [H+] = KW'

which is another way of saying that a solution is acid-base neutral if
the hydrogen ion concentration is equal to the square root of the
KW'. The definitions of acid (acidic) and alkaline (basic)
solutions follow accordingly:

A solution is acidic if [H+] > ROOT (KW')

A solution is basic if [H+] < ROOT (KW')

where "ROOT" is just our way of saying that we are taking a square root.

2. Strong ions in Water

Take some water. Add strong electrolytes, such as NaOH and HCl, which
we know will almost completely dissociate. We now have a heady mix of
water, Na+, Cl-, H+ and OH-
ions. What will happen to the hydrogen ion concentration in this mix?

We already know that water dissociation constrains us to:

[H+] * [OH-] = KW'
.. Equation #0

and we can readily deduce that electrical neutrality ensures that:

[H+] - [OH-] + [Na+] - [Cl-]
= 0 .. Equation #1

We can therefore substitute KW'/[H+]
for [OH-] in Equation #1, and determine [H+].
We get an equation:

[H+] - KW'/[H+] + [Na+] - [Cl-]
= 0

which we can readily tweak by multiplying throughout
by [H+]:

[H+]2
+ [H+] ( [Na+] - [Cl-] )
- KW'
= 0

At last! A use for quadratic equations!
The above is a standard quadratic equation of the form:

a*x2 + b*x + c = 0

We know from our high school days that the solution of such an equation is given by:1

x = - b/2a +- ROOT ( (b/2a)2 - c/a )

Substituting in for
a=1
b= [Na+] - [Cl-]
and c = - KW'
this gives us the formula:

[H+] =
-
([Na+] - [Cl-])/2
+ ROOT (
([Na+] - [Cl-])2/4
+ KW'
)

If we know the amount of sodium and chloride ion in solution, we can
readily determine the hydrogen ion concentration. Even more elegantly,
we can determine the result for any solution containing only strong ions!
We simply fill in the concentrations of those ions in solution, where
we had Na and Cl! The perceptive reader will realise that it is only
the difference in ionic concentrations that matter - we can
abbreviate the above equation to:

[H+] = ROOT ( KW' +
SID2/4) -
SID/2
.. Equation #2

where SID is our abbreviation for the difference between the
concentrations of the strong base cations (eg Na+) and
the strong acid anions (eg Cl-). SID is what Stewart calls
the Strong Ion Difference .. an extremely useful concept. For
a detailed consideration of this elegant solution, see Stewart's book.
Similarly,

[OH-] = ROOT ( KW' +
SID2/4) +
SID/2
.. Equation #3

Note that if SID is negative, then the hydrogen ion concentration is
always greater than the hydroxyl ion concentration. The converse holds
for a positive SID. In these solutions it is clear that if the hydrogen ion
concentration changes the SID must have changed.

If SID is positive and bigger than about 10-6 Eq/l you can see
that KW becomes insignificant, and Equation #3 becomes very
nearly the same as:

[OH-] = [SID]

We can use this and Equation #0 to derive the hydrogen ion concentration:

[H+] = KW' / [SID]

If the SID is negative and bigger than about 10-6 Eq/l then Equation #2 simplifies
out to:
[H+] = -[SID]
but such solutions are not commonly encountered in biological systems.

"Okay, this is all very well" you say "but what is the practical
significance of all this maths?" Easy. The above tells us that
in a solution containing strong ions, if you want to calculate the pH,
you must:

Know the concentrations of the strong ions, and

Plug these values into the equations;

before you can work anything out. And if you add basic or acidic
substances, you cannot just say "We added so much sodium hydroxide so
the pH will change by so much". You have to work things out using the
above equations. Conversely, if you know the values, they are sufficient
to work out the pH.

You can also see that if we have (for example) an acidic solution and
we progressively add base, there will be a sudden, rapid rise in pH
as we approach the point where [SID] is zero, and then add just a tiny bit
more base. Consider the following diagram (our applet) showing how
pH changes with SID - Press the button!

pH is easily measured so now we finally understand the
concept of "titratable acidity" of for example urine (we add base until
there is a sudden rise in pH and this is the point where "titration is
complete" - that is, the SID of the solution is zero). But remember that
this doesn't tell us what titratable acidity means, and also note
that the "sudden change in pH" is an artefact induced because pH is
logarithmic: nothing that dramatic happens if we look at actual hydrogen
ion concentration! Try this - in the above applet, change the
Y axis parameter from pH to [H+] and press on the
Calculate button again!

It is instructive to read Stewart's book
where he looks at titration of interstitial pH using HCl - the
counter-intuitive nature of pH is clearly seen.

Gaps & Gamblegrams

Electrical neutrality in solution demands that Equation #1 is satisfied -
the sum of [negative] and [positive] ions is always zero. This is
conveniently represented by two adjacent bar plots as shown in the
following illustration (after Stewart, p43).

The solution is a simple one containing [H+], [OH-],
[Na+] and [Cl-]. The bar plot (Gamblegram)
clearly shows the disposition of ions.
The minuscule amount of H+ ion is not shown. Such plots can
be used for far more complex solutions, often with unmeasured ions (where
the term "gap" is often used, as in "anion gap"). It is crystal clear from the plot that
the SID is equal to the amount of hydroxyl ion. In alkaline solutions the
SID will be positive, in acidic, negative.

3. A more complex setup - Adding a weak electrolyte

Adding a weak electrolyte (one that only partially dissociates in the
pH range we are considering) complicates things rather a lot. In body
fluids such as plasma, the most important weak electrolyte is albumin,
but the principles hold for all weak electrolytes.

A program to do this

Many people are terrified of computers - an unnecessary state of mind,
as all computers do is give us a way of implementing an algorithm like
the one above in a fast and accurate way!
In order to implement our algorithm we need to express it in
an unambiguous way. We use statements written in a computer
language to do so.

We will now 'formalise' the above algorithm. We won't yet write
it in a specific computer language, we will just make things more
precise, without (we hope) compromising readability. Our only
assumptions are:

We have something (a function) called ROOT
that somehow works out a square root

The function
F(MY_GUESS)
works out Equation #6

the function ABS works out the absolute
value of a number. That is, ABS(-0.001) and ABS(0.001) give the same result,
0.001.

TEA BREAK!

If you were unwise enough to read the above in one sitting, we
suggest that you take a tea break to marshal your resources!

Remember the fundamentals:

There are complex and often non-intuitive relationships
between the various components involved in acid-base.

We must decide what are dependent variables and
which ones are independent.

If we alter an independent variable, then the dependent
variables will change.

If we observe a change in a dependent variable (such as
[H+] or pH) then there must have been a
change in at least one of the independent variables.

4. Strong ions with Carbon dioxide

People who are familiar with acid base as it is commonly taught, usually
cut their teeth on the Henderson-Hasselbalch equation. There is little
wrong with the H-H equation, other than the fact that it only represents
part of the truth. Here we start to explore how carbon dioxide
behaves, but in what we regard as the proper context. Much of the
following may seem familiar, but be careful - don't lose sight of
the big picture!

Take our familiar mixture of strong ions and water, and
expose it to CO2. What happens?
Four things can happen to CO2 gas when exposed to water -
it can dissolve, react with water to form carbonic acid, or even form
bicarbonate or carbonate ions. We will explore each of these in turn, but
the two most significant reactions are the formation of carbonate and
bicarbonate, as each has its own equilibrium constant. By now you will
realise that these reactions with their equilibrium constants will have
a profound influence on the whole system, and it is only
in the context of the whole system that we can understand the
role of carbon dioxide. Let's see:

CO2 can dissolve, as expressed by the equation:

CO2(gas) <=====> CO2(dissolved)

Forward Reaction

Reverse Reaction

===> Depends on partial pressure of CO2

<=== Depends on concentration of dissolved CO2

Rate of forward reaction = Kf * PCO2

Rate of reverse reaction = Kr * [CO2(dissolved)]

..AT EQUILIBRIUM..

[CO2(dissolved)]
= SCO2 * PCO2
.. Equation #7A

For Kf/Kr we have substituted
SCO2, otherwise called the Solubility of CO2.
In other words, the amount of dissolved CO2 depends
on the partial pressure of CO2 times a rate constant.
SCO2 is dependent on temperature, and at 37 C it
is about 3.0 * 10-5 Eq/litre/mmHg.
Notethat if we
are examining a fluid that isn't in contact with gas, we still
talk about a partial pressure of that gas in solution "as if"
it were exposed to, and in equilibrium with, a gas containing
that gas at that partial pressure. This is just another convenient
way of representing the concentration of dissolved gas in solution.

CO2 can react with water to form carbonic acid:

CO2 + H2O <==> H2CO3

Equilibrium is represented by:

[CO2(dissolved)]
* [h30] = K * [H2CO3]
.. Equation #7B

If we treat [h30] as constant, and rearrange things a bit we get:

[H2CO3] = KH * PCO2

The value of KH at 37 C is 9 * 10-8 Eq/litre -
because of this, the H2CO3 concentration is
far smaller than the amount of dissolved CO2.

The reaction of CO2 with water is
SLOW , with a half time of about 30 seconds, fortunately
speeded up to microseconds by the carbonic anhydrase abundantly present
in most tissues.

H2CO3 thus formed can dissociate into
bicarbonate and hydrogen ions:

H2CO3 <==> H+ + HCO3-

Equilibrium is represented by:

[H+] * [HCO3-] = K * [H2CO3]

It follows that:

[H+] * [HCO3-] =
KC * PCO2
.. Equation #8

A good value for KC is 2.6 * 10-11
(Eq/l)2/mmHg.

Once formed, HCO3- may rapidly
dissociate:

HCO3- <==> H+ + CO32-

Equilibrium is represented by:

[H+] * [CO32-] =
K3 * [HCO3-]
.. Equation #9

A typical value for K3 is 6 * 10-11 Eq/litre.

The big picture - finding the pH!

As usual, we need four simultaneous equations to work out all the dependent
variables (given the independent ones). These are our old familiar equation
#0, Equations #8 and #9, and the requirement for electrical neutrality:

[H+] * [OH-] = KW'
.. Equation #0

[H+] * [HCO3-] =
KC * PCO2
.. Equation #8

[H+] * [CO32-] =
K3 * [HCO3-]
.. Equation #9

[SID] + [H+] - [OH-] - [HCO3-]
- [CO32-] = 0
.. Equation #10

It is important to distinguish between dependent and independent
variables. Bicarbonate concentration and hydrogen ion concentration
are dependent variables, but SID and pCO2 are independent!

We work out our equation for [H+] in the same way we
did in Section 3 above, and solve it using a computer, again as above.
On contemplating the results thus obtained, a whole host of interesting
conclusions emerge. These include:

Where the SID is negative, the excess negative strong ions must
be balanced by positive ions (H+ being the only candidate)
so OH-, HCO3- and
HCO32- become rare and endangered species -
their concentration becomes negligible.

With a positive SID, [HCO3-] very accurately
approximates SID. This equality is immensely useful, and
also implies that [HCO3-] is independent
of PCO2 in interstitial fluid!
(at least in the physiological range).

Totally counter-intuitive
is that [CO32-]
decreases with increasing PCO2!!
Conversely, if we lower the PCO2 enough, carbonate levels,
although tiny, may rise to a point where the solubility
product for calcium carbonate (10-8) is exceeded.
This may lower calcium levels, causing tetany!
Click on the calculate button below, to see how
carbonate concentration increases as PCO2 decreases:

and a Weak Acid

The combination of strong ions, carbon dioxide and a weak acid
closely models blood plasma, but also provides a fairly
accurate representation of intracellular fluids. Blood plasma is rich
in weak acids, the majority being proteins. For the purposes of analysis
it is probably moderately accurate to regard them as being all one acid
with a single ATOT and single KA. This assumption
is not however central to Stewart's work - as is seen in the articles
by Figge et al who extend his model with multiple
Ka's.

Following the pattern we established in previous sections, we
identify the independent variables ( [SID],
PCO2, ATOT which
are respectively the strong ion difference, partial pressure of CO2
and the total amount of weak acid present). In addition, we need to know
KW', KA, KC and K3. Given these,
we can calculate any one of eight dependent variables:
HCO3-, A-, HA, CO2(dissolved),
CO32-, H2CO3, OH-,
and H+. Note that dissolved CO2 and H2CO3
are easily determined from Equations #7A and #7B.

Exactly as we have done before, we derive six simultaneous equations,
most of which are old friends:

Thanks too to PJ Hilton, who pointed out a typo in Stewart's book, which we carried
over in a mildly embarrassing fashion.

Recent developments are explored on
yet another page on anaesthetist.com.
Further constructive comment is of course still welcome. By the way the actual
distance between Brown University
(Stewart) and Copenhagen (Siggaard-Andersen) is 5958 km, just for the
record.

References

Stewart PA How to Understand Acid-Base. A Quantitative
Acid-Base Primer for Biology and Medicine 1981 Edward Arnold.
ISBN 0-7131-4390-8. In our opinion, this book is pure gold.
Get a copy!

It's a tragedy that Peter Stewart is dead - the
man was a genius}

Stewart PA Can J Physiol Pharmacol 1983 61 1444
A good overview of modern acid-base.